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Problem

[[0.30,0.20,0.30],[0.40,0.30,0.30],[0.30,0.30,0.20]]−[[1,0,0],[0,1,0],[0,0,1]]*[[81],[8],[9]]

Solution

1. Identify the identity matrix I and the vector v Multiplying any vector by the identity matrix I results in the original vector.

[[1,0,0],[0,1,0],[0,0,1]]*[[81],[8],[9]]=[[81],[8],[9]]

2. Recognize that the operation involves a 3×3 matrix and a 3×1 vector. Matrix subtraction is only defined for matrices of the same dimensions.

3. Interpret the task as a matrix-vector operation where the matrix A acts on the vector v followed by the subtraction of the second term.

A*v−I*v

4. Multiply the first matrix by the vector.

[[0.30,0.20,0.30],[0.40,0.30,0.30],[0.30,0.30,0.20]]*[[81],[8],[9]]=[[0.30*(81)+0.20*(8)+0.30*(9)],[0.40*(81)+0.30*(8)+0.30*(9)],[0.30*(81)+0.30*(8)+0.20*(9)]]

5. Calculate the components of the resulting vector.

[[24.3+1.6+2.7],[32.4+2.4+2.7],[24.3+2.4+1.8]]=[[28.6],[37.5],[28.5]]

6. Subtract the vector obtained in step 1 from the vector obtained in step 5.

[[28.6],[37.5],[28.5]]−[[81],[8],[9]]=[[28.6−81],[37.5−8],[28.5−9]]

7. Simplify the final values.

[[−52.4],[29.5],[19.5]]

Final Answer

[[0.30,0.20,0.30],[0.40,0.30,0.30],[0.30,0.30,0.20]]*[[81],[8],[9]]−[[1,0,0],[0,1,0],[0,0,1]]*[[81],[8],[9]]=[[−52.4],[29.5],[19.5]]

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